3) The answer comes to us from Josephus. In "Antiquities of the Jews" the historian tells us the Old Testament in his own words, using as his sources books that were salvaged from Solomon's Temple at the time of the Roman-Jewish war. His work on the "Divided Kingdom" (beginning with Jeroboam and Rehoboam and ending with the fall of Judah) can be found in Books 8, 9 and 10 of "Antiquities" (refs at the end of this paper).
In William Whiston's translation of Josephus' work (Antiquities / Wars / Apion / Life of Josephus), Whiston adds an essay of his own, making a very strong case for believing that Josephus' sources were actually older (written in an older Hebrew script) and more dependable than our modern Old Testament text. Josephus' work often differs from the Bible we know today, but there is a surprise waiting.
Josephus is in almost perfect harmony with our modern Bible king data. In his Northern king list, all the reign lengths are identical to the modern Bible, with two exceptions being Jehu and Jeroboam II, each of whom are given only one year less than the Bible gives.
On the Southern list, all the reign lengths are identical to the Bible. So, in total, of thirty nine reign lengths (for all the Northern and Southern kings), all are exactly as we know them except for two, which are each only one year shorter than their Biblical counterparts.
The Bible gives specific ages for sixteen of the Southern kings (no such info is given for the Northern kings), at the time each came to power. Josephus tells fourteen of these ages and they all match the Bible exactly. (The ages have to be calculated. Josephus tells us the age of the king when he died and the number of years he reigned.)
The Bible provides a total of twenty-nine cross references for the Northern and Southern lists. Josephus employs a total of only nine cross references, but our answer is buried here. Josephus' Northern list contains just five cross references. For Nadab, Jehoash and Jeroboam II the references are identical to the Bible. For Omri the reference differs by just one year, and for Jehoahaz the reference differs by only two years.
Josephus' Southern list contains just four cross references. For Abijam and Amaziah the references are identical to the Bible. For Hezekiah the reference differs by just one year. We'll come back to the fourth cross reference in a moment.
Josephus notes that Hoshea's 9th year was the same as Hezekiah's 7th (Ant 9:14:1), differing from the Bible again by only one year (2Kings 18:10 - Hoshea's 9th equals Hezekiah's 6th). And he agrees with the Bible that Sennacherib invaded Jerusalem in Hezekiah's 14th year (Ant 9:10:1 / 2K 18:13).
All Josephus' Northern and Southern reign lengths, all his cross references, and all his Southern kings' ages - approximately sixty-five pieces of information - are consistent enough with the Bible that we can trust their sources as being essentially identical.
There is still one cross reference that we haven't examined. It is for Uzziah, and it differs from the Bible by thirteen years! The Bible said that Uzziah began in Jeroboam's 27th year, and that was the source of our only problem. But Josephus says that Uzziah began his reign in the fourteenth year of Jeroboam (see Antiquities 9:10:3). If we use Josephus' reference, we find that Uzziah has been placed exactly where we want him, even two years earlier, beginning his reign in 806 BCE, two years before his father Amaziah dies. Everything now falls beautifully into place.
Uzziah's new date actually pulls everything back in time two years, so the whole chart has been adjusted forward two years to keep its final date at 586 BCE.